A Unique Decomposition Result for HT Factors with Torsion Free Core

Abstract

We prove that II${}_1$ factors $M$ have a unique (up to unitary conjugacy) cross-product type decomposition around ``core subfactors'' $N\subset M$ satisfying the property HT and a certain ``torsion freeness'' condition. In particular, this shows that isomorphism of factors of the form $L_{\alpha }({\mathbb{Z}}^2)⋊\Gamma$, for torsion free, non-amenable subgroups $\Gamma \subset SL(2,\mathbb{Z})$ and $\alpha =e^{2\pi it}$, $t\notin \mathbb{Q}$, implies isomorphism of the corresponding groups $\Gamma$.

8 pages; minor corrections 02/17/04, 03/17/04